Ro'yxatga olish sanasi 21 Aprel 2025
15 SotishChiziqli algebraik tenglamalar sistemasini zeydel usuli yordamida matlab dasturidan foydalanib yechish
![ko`rinishda yozib olamiz va dastlabki yaqinlashish x sifatida oddiy iteratsiya
usulidagidek x =(0,6; 0,44; 0,95;1; 1,6) deb olamiz.
Iteratsiyaning birinchi kadamini bajaramiz:
Keyingi yaqinlashishlarni 3.5- jadvalda keltiramiz:
3.5-jadval
kх1
(k) х2
(k) х3
(k) х4
(k) х5
(k)
0 0,6 0,44 0,95 1 1,6
1 0.881 0,771 0,937 1,817 1,948
2 0,973 0,961 0,985 1.974 1,992
3 0,995 0,995 0,999 1,996 1,999
4 0,9995 0,9991 0,9997 1,9995 1,9998
5 0,99992 0,99989 0,99997 1.99991 1,99997
6 0,99999 0,99998 0,99999 1,99999 2.00000
Matlab kodi:
% Gauss-Seidel usuli
% Chiziqli tenglamalar sistemasini belgilaymiz:
% A * x = b
A = [4, -1, 0, 0; -1, 4, -1, 0; 0, -1, 4, -1; 0, 0, -1, 3]; b = [15; 10; 10; 10];
% Boshlang'ich taxmin
x0 = zeros(size(b)); % [0; 0; 0; 0]
% Aniqlik uchun talab qilinadigan qiymat
epsilon = 1e-6;
12](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_11.png?v=1)
![Misol 2. x2
(k+1)=
b2
a22
−
a21
a22
x1
(k+1)− ∑
j=3
n a2j
a22
xj
(k)
% Gauss-Seidel usuli
% Chiziqli tenglamalar sistemasini belgilaymiz:
% A * x = b
A = [4, -1, 0, 0;
-1, 4, -1, 0;
0, -1, 4, -1;
0, 0, -1, 3];
b = [15; 10; 10; 10];
% Boshlang'ich taxmin
x0 = zeros(size(b)); % [0; 0; 0; 0]
% Aniqlik uchun talab qilinadigan qiymat
epsilon = 1e-6;
% Maksimal iteratsiya soni
max_iter = 100;
% Gauss-Zeydel iteratsion protsedurasi
n = length(b);
x = x0;
for k = 1:max_iter
x_old = x;
for i = 1:n
if i == 1
% Birinchi o'zgaruvchi x_1
x(i) = (b(i) - sum(A(i, i+1:n) .* x_old(i+1:n))) / A(i, i);
elseif i == 2
% Ikkinchi o'zgaruvchi x_2
sum1 = A(i, 1) * x(1); % x1^(k+1) dan foydalanamiz
sum2 = sum(A(i, i+1:n) .* x_old(i+1:n));
x(i) = (b(i) - sum1 - sum2) / A(i, i);
else
% Qolgan o'zgaruvchilar uchun umumiy formula
sum1 = sum(A(i, 1:i-1) .* x(1:i-1));
sum2 = sum(A(i, i+1:n) .* x_old(i+1:n));
x(i) = (b(i) - sum1 - sum2) / A(i, i);
end
end
% Xatolikni tekshirish
if norm(x - x_old, inf) < epsilon
fprintf('Iteratsiya %d da to'xtadi.
', k);
break;
end
end
% Natija fprintf('Yechim:');
disp(x);
14](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_13.png?v=1)
![Gauss-Zeydel usuli sifatida ham tanilgan Zeydel usuli chiziqli
tenglamalar tizimini iterativ tarzda yechish uchun dasturlash tilida amalga
oshirilishi mumkin.
Pytonda Zeydel usulini amalga oshirish kodi
Input:
A: koeffitsient matritsasi (n x n)
b: o'ng tomonli vektor (n x 1)
x: dastlabki taxmin vektori (n x 1)
max_iterations: takrorlashlarning maksimal soni
tolerance: konvergentsiyaga tolerantlik
Output:
x: yechim vektori (n x 1)
Procedure SeidelMethod(A, b, x, max_iterations, tolerance):
n = number of rows/columns in A
iteration = 0
error = infinity
while iteration < max_iterations and error > tolerance:
iteration = iteration + 1
x_old = copy of x
error = 0
for i = 1 to n do:
sum = 0
for j = 1 to n do:
if j ≠ i then
sum = sum + A[i][j] * x[j]
end for
x[i] = (b[i] - sum) / A[i][i]
error = max(error, abs(x[i] - x_old[i]))
end for
end while return
15](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_14.png?v=1)
![if j ~= i
sum = sum + A(i, j) * x(j);
end
end
% Yangi x[i] qiymatini hisoblash
x(i) = (b(i) - sum) / A(i, i);
% Maksimal xatoni aniqlash
error = max(error, abs(x(i) - x_old(i)));
end
end
% Agar konvergentsiya bo'lmasa, ogohlantirish chiqarish
if iteration == max_iterations && error > tolerance
warning('Gauss-Seidel usuli konvergentsiyaga erishmadi. Maksimal iteratsiyaga yetildi.');
end
end
Ushbu Python misoli matritsa operatsiyalari uchun NumPy yordamida Zeydel
usulini qanday amalga oshirishni ko'rsatadi. Kirish matritsalarini sozlang A va b,
maksimal takrorlashlar soni va tolerantlik bilan bir qatorda, muayyan muammo
talablariga mos keladi.
18](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_17.png?v=1)
![C++ da Zeydel usulini amalga oshirishning qisqacha misoli:
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
vector<double> seidel_method(const vector<vector<double>>& A, const vector<double>& b,
vector<double> x, int max_iterations, double tolerance) {
int n = b.size();
vector<double> x_old(n);
double error = tolerance + 1;
int iteration = 0;
while (iteration < max_iterations && error > tolerance) {
x_old = x;
error = 0.0;
for (int i = 0; i < n; ++i) {
double sum = 0.0;
for (int j = 0; j < n; ++j) {
if (j != i) {
sum += A[i][j] * x[j];
}
}
x[i] = (b[i] - sum) / A[i][i];
error = max(error, abs(x[i] - x_old[i]));
}
iteration++;
}
return x;
}
int main() {
vector<vector<double>> A = {{10, 2, 1}, {1, 5, 1}, {2, 3, 10}};
vector<double> b = {7, -8, 6};
vector<double> x(b.size(), 0);
int max_iterations = 100;
double tolerance = 1e-6;
vector<double> solution = seidel_method(A, b, x, max_iterations, tolerance);
cout << "Solution vector:" << endl;
for (int i = 0; i < solution.size(); ++i) {
cout << "x[" << i << "] = " << solution[i] << endl;
}
return 0;
}
qisqacha mazmuni:
19](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_18.png?v=1)
![MATLAB da amalga oshirish qisqacha misol:
function x = gauss_seidel(A, b, x0, tol, max_iterations)
if nargin < 4
tol = 1e-10;
end
if nargin < 5
max_iterations = 1000;
end
n = length(b);
x = x0;
for k = 1:max_iterations
x_old = x;
for i = 1:n
sum1 = sum(A(i, 1:i-1) .* x(1:i-1)');
sum2 = sum(A(i, i+1:n) .* x_old(i+1:n)');
x(i) = (b(i) - sum1 - sum2) / A(i, i);
end
if norm(x - x_old, inf) < tol
fprintf('Converged in %d iterations\n', k);
return;
end
end
fprintf('Maximum iterations reached\n');
end
A = [4, -1, 0, 0; -1, 4, -1, 0; 0, -1, 4, -1; 0, 0, -1, 3];
b = [15; 10; 10; 10];
x0 = zeros(size(b));
x = gauss_seidel(A, b, x0)
disp('Solution:')
21](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_20.png?v=1)
![ Turli sharoitlarda usulning aniqligi va samaradorligini baholash uchun
raqamli ko'rsatkichlar va test holatlaridan foydalaning.
Matlab tilidagi Zeydel usulining takomillashtirilgan namunasi:
function x = seidel_method(A, b, x, max_iterations, tolerance, omega)
n = length(b);
x_old = zeros(n, 1);
error = tolerance + 1;
iteration = 0;
while iteration < max_iterations && error > tolerance
x_old = x;
error = 0;
for i = 1:n
sum = 0;
for j = 1:n
if j ~= i
sum = sum + A(i,j) * x(j);
end
end
x(i) = (1 - omega) * x(i) + (omega / A(i,i)) * (b(i) - sum);
error = max(error, abs(x(i) - x_old(i)));
end
iteration = iteration + 1;
end
end
% Example usage
A = [10 2 1; 1 5 1; 2 3 10];
b = [7; -8; 6];
x = zeros(3, 1);
max_iterations = 100;
tolerance = 1e-6;
omega = 1.2;
solution = seidel_method(A, b, x, max_iterations, tolerance, omega);
29](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_28.png?v=1)
![fprintf('Solution vector:\n');
for i = 1:length(solution)
fprintf('x[%d] = %f\n', i, solution(i));
end
Zeydel usulida (yoki har qanday iterativ usulda) samaradorlikni oshirish va
xatolarni bartaraf etish algoritmik takomillashtirish, ehtiyotkorlik bilan dasturlash
amaliyoti va raqamli jihatlarni hisobga olish kombinatsiyasini o'z ichiga oladi.
Ushbu strategiyalarni qo'llash orqali siz keng ko'lamli chiziqli tizimni hal qilish
vazifalarini yaxshiroq hal qilish uchun amalga oshirishning ishlashi, ishonchliligi
va aniqligini optimallashtirishingiz mumkin.
Zeydel usulining ko'p qirraliligi va samaradorligi uni turli ilmiy, muhandislik
va hisoblash fanlari bo'yicha qimmatli vositaga aylantiradi. Uning keng miqyosli
chiziqli tizimlarni iterativ tarzda boshqarish qobiliyati uni to'g'ridan-to'g'ri usullar
amaliy bo'lmagan yoki hisoblash qimmat bo'lishi mumkin bo'lgan ilovalar uchun
mos qiladi. Zeydel usuli kabi iterativ usullardan foydalangan holda tadqiqotchilar
va muhandislar murakkab muammolarni yanada samaraliroq hal qilishlari
mumkin, bu esa texnologiya, fan va sanoatdagi yutuqlarga olib keladi.
30](https://docx.uz/documents/89d93666-d47c-42b6-bcfb-06e398891e46/page_29.png?v=1)
Chiziqli algebraik tenglamalar sistemasini zeydel usuli yordamida matlab dasturidan foydalanib yechish